\documentclass[9pt]{article}
\usepackage{amsmath}
\begin{document}
Read in the following dictionary:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{6}   &  17.0 & -1.00 x_{1} & -8.00 x_{2} & +  3.00 x_{3} & -7.00 x_{4} &   \\
 x_{7}   &  -3.0 & +  4.00 x_{1} & +  7.00 x_{2} & +  9.00 x_{3} & -10.00 x_{4} & -4.00 x_{5}\\
 x_{8}   &  31.0 & -7.00 x_{1} & -9.00 x_{2} & +  4.00 x_{3} &   & +  6.00 x_{5}\\
 x_{9}   &  5.0 & -9.00 x_{1} & +  7.00 x_{2} & +  5.00 x_{3} &   & -1.00 x_{5}\\
 x_{10}   &  0.0 & -9.00 x_{1} & +  8.00 x_{2} & +  8.00 x_{3} & +  1.00 x_{4} & +  8.00 x_{5}\\
 x_{11}   &  37.0 & -7.00 x_{1} & -8.00 x_{2} & +  2.00 x_{3} & +  5.00 x_{4} & -5.00 x_{5}\\
 x_{12}   &  -2.0 & +  3.00 x_{1} & +  9.00 x_{2} & -2.00 x_{3} & +  1.00 x_{4} & + 10.00 x_{5}\\
 x_{13}   &  -40.0 & +  7.00 x_{1} & +  5.00 x_{2} & + 10.00 x_{3} &   & + 10.00 x_{5}\\
 x_{14}   &  10.0 & +  3.00 x_{1} & -6.00 x_{2} & -3.00 x_{3} & +  5.00 x_{4} & -10.00 x_{5}\\
\hline
z    &  0.0 & -4.00 x_{1} & +  3.00 x_{2} & +  2.00 x_{3} & -3.00 x_{4} & +  2.00 x_{5}\\
\end{array}\]
\subsection{Initialization Phase: Dual Problem Solving}
New Objective in primal was changed to : \[ \max\ \sum_{j=1}^{5}\ - x_j \] 
Primal variable $x_j$ corresponds to dual variable $y_j$ for $j = 1,\ldots,14$
Dual Dictionary (with objective changed is): 
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{1}   &  1.0 & +  1.00 y_{6} & -4.00 y_{7} & +  7.00 y_{8} & +  9.00 y_{9} & +  9.00 y_{10} & +  7.00 y_{11} & -3.00 y_{12} & -7.00 y_{13} & -3.00 y_{14}\\
 y_{2}   &  1.0 & +  8.00 y_{6} & -7.00 y_{7} & +  9.00 y_{8} & -7.00 y_{9} & -8.00 y_{10} & +  8.00 y_{11} & -9.00 y_{12} & -5.00 y_{13} & +  6.00 y_{14}\\
 y_{3}   &  1.0 & -3.00 y_{6} & -9.00 y_{7} & -4.00 y_{8} & -5.00 y_{9} & -8.00 y_{10} & -2.00 y_{11} & +  2.00 y_{12} & -10.00 y_{13} & +  3.00 y_{14}\\
 y_{4}   &  1.0 & +  7.00 y_{6} & + 10.00 y_{7} &    &   & -1.00 y_{10} & -5.00 y_{11} & -1.00 y_{12} &   & -5.00 y_{14}\\
 y_{5}   &  1.0  &   & +  4.00 y_{7} & -6.00 y_{8} & +  1.00 y_{9} & -8.00 y_{10} & +  5.00 y_{11} & -10.00 y_{12} & -10.00 y_{13} & + 10.00 y_{14}\\
\hline
z    &  -0 & -17.00 y_{6} & +  3.00 y_{7} & -31.00 y_{8} & -5.00 y_{9} &   & -37.00 y_{11} & +  2.00 y_{12} & + 40.00 y_{13} & -10.00 y_{14}\\
\end{array}\]
Initialization succeeded in finding final dual dictionary with 6 pivots
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{14}   &  0.0391304347826 & +  0.26 y_{6} & +  0.04 y_{3} & +  1.37 y_{8} & +  1.34 y_{9} & +  1.90 y_{10} & +  0.76 y_{11} & -0.32 y_{7} & +  0.05 y_{5} & -0.13 y_{1}\\
 y_{12}   &  0.0228260869565 & +  0.40 y_{6} & +  0.11 y_{3} & +  0.63 y_{8} & +  1.28 y_{9} & +  1.11 y_{10} & +  1.03 y_{11} & +  0.90 y_{7} & -0.06 y_{5} & -0.08 y_{1}\\
 y_{2}   &  0.447826086957 & +  6.65 y_{6} & -0.39 y_{3} & + 10.83 y_{8} & -11.29 y_{9} & -6.54 y_{10} & +  2.15 y_{11} & -12.90 y_{7} & +  0.77 y_{5} & +  0.17 y_{1}\\
 y_{4}   &  0.78152173913 & +  5.29 y_{6} & -0.33 y_{3} & -7.50 y_{8} & -8.00 y_{9} & -11.63 y_{10} & -9.83 y_{11} & + 10.71 y_{7} & -0.18 y_{5} & +  0.73 y_{1}\\
 y_{13}   &  0.116304347826 & -0.14 y_{6} & -0.07 y_{3} & +  0.14 y_{8} & +  0.16 y_{9} & -0.01 y_{10} & +  0.23 y_{11} & -0.82 y_{7} & +  0.00 y_{5} & -0.05 y_{1}\\
\hline
z    &  4.30652173913 & -24.46 y_{6} & -2.83 y_{3} & -37.90 y_{8} & -9.48 y_{9} & -17.08 y_{10} & -33.21 y_{11} & -24.69 y_{7} & -0.46 y_{5} & -1.02 y_{1}\\
\end{array}\]
Primal Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{6}   &  24.4565217391 & -0.26 x_{14} & -0.40 x_{12} & -6.65 x_{2} & -5.29 x_{4} & +  0.14 x_{13}\\
 x_{3}   &  2.82608695652 & -0.04 x_{14} & -0.11 x_{12} & +  0.39 x_{2} & +  0.33 x_{4} & +  0.07 x_{13}\\
 x_{8}   &  37.9043478261 & -1.37 x_{14} & -0.63 x_{12} & -10.83 x_{2} & +  7.50 x_{4} & -0.14 x_{13}\\
 x_{9}   &  9.47608695652 & -1.34 x_{14} & -1.28 x_{12} & + 11.29 x_{2} & +  8.00 x_{4} & -0.16 x_{13}\\
 x_{10}   &  17.0826086957 & -1.90 x_{14} & -1.11 x_{12} & +  6.54 x_{2} & + 11.63 x_{4} & +  0.01 x_{13}\\
 x_{11}   &  33.2065217391 & -0.76 x_{14} & -1.03 x_{12} & -2.15 x_{2} & +  9.83 x_{4} & -0.23 x_{13}\\
 x_{7}   &  24.6869565217 & +  0.32 x_{14} & -0.90 x_{12} & + 12.90 x_{2} & -10.71 x_{4} & +  0.82 x_{13}\\
 x_{5}   &  0.458695652174 & -0.05 x_{14} & +  0.06 x_{12} & -0.77 x_{2} & +  0.18 x_{4} & -0.00 x_{13}\\
 x_{1}   &  1.02173913043 & +  0.13 x_{14} & +  0.08 x_{12} & -0.17 x_{2} & -0.73 x_{4} & +  0.05 x_{13}\\
\hline
z    &  -4.30652173913 & -0.04 x_{14} & -0.02 x_{12} & -0.45 x_{2} & -0.78 x_{4} & -0.12 x_{13}\\
\end{array}\]
Primal Dictionary with original objective is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{6}   &  24.4565217391 & -0.26 x_{14} & -0.40 x_{12} & -6.65 x_{2} & -5.29 x_{4} & +  0.14 x_{13}\\
 x_{3}   &  2.82608695652 & -0.04 x_{14} & -0.11 x_{12} & +  0.39 x_{2} & +  0.33 x_{4} & +  0.07 x_{13}\\
 x_{8}   &  37.9043478261 & -1.37 x_{14} & -0.63 x_{12} & -10.83 x_{2} & +  7.50 x_{4} & -0.14 x_{13}\\
 x_{9}   &  9.47608695652 & -1.34 x_{14} & -1.28 x_{12} & + 11.29 x_{2} & +  8.00 x_{4} & -0.16 x_{13}\\
 x_{10}   &  17.0826086957 & -1.90 x_{14} & -1.11 x_{12} & +  6.54 x_{2} & + 11.63 x_{4} & +  0.01 x_{13}\\
 x_{11}   &  33.2065217391 & -0.76 x_{14} & -1.03 x_{12} & -2.15 x_{2} & +  9.83 x_{4} & -0.23 x_{13}\\
 x_{7}   &  24.6869565217 & +  0.32 x_{14} & -0.90 x_{12} & + 12.90 x_{2} & -10.71 x_{4} & +  0.82 x_{13}\\
 x_{5}   &  0.458695652174 & -0.05 x_{14} & +  0.06 x_{12} & -0.77 x_{2} & +  0.18 x_{4} & -0.00 x_{13}\\
 x_{1}   &  1.02173913043 & +  0.13 x_{14} & +  0.08 x_{12} & -0.17 x_{2} & -0.73 x_{4} & +  0.05 x_{13}\\
\hline
z    &  2.48260869565 & -0.70 x_{14} & -0.41 x_{12} & +  2.94 x_{2} & +  0.93 x_{4} & -0.09 x_{13}\\
\end{array}\]
\section{Optimization Phase Simplex}
Starting Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{6}   &  24.4565217391 & -0.26 x_{14} & -0.40 x_{12} & -6.65 x_{2} & -5.29 x_{4} & +  0.14 x_{13}\\
 x_{3}   &  2.82608695652 & -0.04 x_{14} & -0.11 x_{12} & +  0.39 x_{2} & +  0.33 x_{4} & +  0.07 x_{13}\\
 x_{8}   &  37.9043478261 & -1.37 x_{14} & -0.63 x_{12} & -10.83 x_{2} & +  7.50 x_{4} & -0.14 x_{13}\\
 x_{9}   &  9.47608695652 & -1.34 x_{14} & -1.28 x_{12} & + 11.29 x_{2} & +  8.00 x_{4} & -0.16 x_{13}\\
 x_{10}   &  17.0826086957 & -1.90 x_{14} & -1.11 x_{12} & +  6.54 x_{2} & + 11.63 x_{4} & +  0.01 x_{13}\\
 x_{11}   &  33.2065217391 & -0.76 x_{14} & -1.03 x_{12} & -2.15 x_{2} & +  9.83 x_{4} & -0.23 x_{13}\\
 x_{7}   &  24.6869565217 & +  0.32 x_{14} & -0.90 x_{12} & + 12.90 x_{2} & -10.71 x_{4} & +  0.82 x_{13}\\
 x_{5}   &  0.458695652174 & -0.05 x_{14} & +  0.06 x_{12} & -0.77 x_{2} & +  0.18 x_{4} & -0.00 x_{13}\\
 x_{1}   &  1.02173913043 & +  0.13 x_{14} & +  0.08 x_{12} & -0.17 x_{2} & -0.73 x_{4} & +  0.05 x_{13}\\
\hline
z    &  2.48260869565 & -0.70 x_{14} & -0.41 x_{12} & +  2.94 x_{2} & +  0.93 x_{4} & -0.09 x_{13}\\
\end{array}\]


 $ x_{2} $ enters and $ x_{5} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{6}   &  20.4915254237 & +  0.15 x_{14} & -0.88 x_{12} & +  8.64 x_{5} & -6.88 x_{4} & +  0.17 x_{13}\\
 x_{3}   &  3.0593220339 & -0.07 x_{14} & -0.08 x_{12} & -0.51 x_{5} & +  0.42 x_{4} & +  0.06 x_{13}\\
 x_{8}   &  31.4463276836 & -0.70 x_{14} & -1.42 x_{12} & + 14.08 x_{5} & +  4.92 x_{4} & -0.09 x_{13}\\
 x_{9}   &  16.2062146893 & -2.05 x_{14} & -0.47 x_{12} & -14.67 x_{5} & + 10.70 x_{4} & -0.21 x_{13}\\
 x_{10}   &  20.9802259887 & -2.31 x_{14} & -0.64 x_{12} & -8.50 x_{5} & + 13.19 x_{4} & -0.02 x_{13}\\
 x_{11}   &  31.9237288136 & -0.63 x_{14} & -1.18 x_{12} & +  2.80 x_{5} & +  9.32 x_{4} & -0.22 x_{13}\\
 x_{7}   &  32.3785310734 & -0.48 x_{14} & +  0.03 x_{12} & -16.77 x_{5} & -7.63 x_{4} & +  0.76 x_{13}\\
 x_{2}   &  0.59604519774 & -0.06 x_{14} & +  0.07 x_{12} & -1.30 x_{5} & +  0.24 x_{4} & -0.00 x_{13}\\
 x_{1}   &  0.918079096045 & +  0.14 x_{14} & +  0.06 x_{12} & +  0.23 x_{5} & -0.77 x_{4} & +  0.06 x_{13}\\
\hline
z    &  4.23446327684 & -0.89 x_{14} & -0.20 x_{12} & -3.82 x_{5} & +  1.63 x_{4} & -0.11 x_{13}\\
\end{array}\]


 $ x_{4} $ enters and $ x_{1} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{6}   &  12.2844036697 & -1.11 x_{14} & -1.45 x_{12} & +  6.62 x_{5} & +  8.94 x_{1} & -0.32 x_{13}\\
 x_{3}   &  3.55963302752 & +  0.01 x_{14} & -0.05 x_{12} & -0.39 x_{5} & -0.54 x_{1} & +  0.09 x_{13}\\
 x_{8}   &  37.3119266055 & +  0.20 x_{14} & -1.01 x_{12} & + 15.52 x_{5} & -6.39 x_{1} & +  0.26 x_{13}\\
 x_{9}   &  28.9633027523 & -0.08 x_{14} & +  0.41 x_{12} & -11.53 x_{5} & -13.90 x_{1} & +  0.56 x_{13}\\
 x_{10}   &  36.7155963303 & +  0.11 x_{14} & +  0.45 x_{12} & -4.62 x_{5} & -17.14 x_{1} & +  0.92 x_{13}\\
 x_{11}   &  43.0366972477 & +  1.08 x_{14} & -0.41 x_{12} & +  5.53 x_{5} & -12.10 x_{1} & +  0.44 x_{13}\\
 x_{7}   &  23.2752293578 & -1.88 x_{14} & -0.60 x_{12} & -19.01 x_{5} & +  9.92 x_{1} & +  0.22 x_{13}\\
 x_{2}   &  0.880733944954 & -0.02 x_{14} & +  0.09 x_{12} & -1.23 x_{5} & -0.31 x_{1} & +  0.01 x_{13}\\
 x_{4}   &  1.19266055046 & +  0.18 x_{14} & +  0.08 x_{12} & +  0.29 x_{5} & -1.30 x_{1} & +  0.07 x_{13}\\
\hline
z    &  6.18348623853 & -0.59 x_{14} & -0.06 x_{12} & -3.34 x_{5} & -2.12 x_{1} & +  0.01 x_{13}\\
\end{array}\]


 $ x_{13} $ enters and $ x_{6} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{13}   &  38.0397727273 & -3.44 x_{14} & -4.49 x_{12} & + 20.51 x_{5} & + 27.68 x_{1} & -3.10 x_{6}\\
 x_{3}   &  7.11931818182 & -0.31 x_{14} & -0.47 x_{12} & +  1.53 x_{5} & +  2.05 x_{1} & -0.29 x_{6}\\
 x_{8}   &  47.1534090909 & -0.69 x_{14} & -2.17 x_{12} & + 20.83 x_{5} & +  0.77 x_{1} & -0.80 x_{6}\\
 x_{9}   &  50.1818181818 & -2.00 x_{14} & -2.09 x_{12} & -0.09 x_{5} & +  1.55 x_{1} & -1.73 x_{6}\\
 x_{10}   &  71.8238636364 & -3.06 x_{14} & -3.69 x_{12} & + 14.31 x_{5} & +  8.41 x_{1} & -2.86 x_{6}\\
 x_{11}   &  59.8579545455 & -0.44 x_{14} & -2.40 x_{12} & + 14.60 x_{5} & +  0.14 x_{1} & -1.37 x_{6}\\
 x_{7}   &  31.5113636364 & -2.63 x_{14} & -1.57 x_{12} & -14.57 x_{5} & + 15.91 x_{1} & -0.67 x_{6}\\
 x_{2}   &  1.36931818182 & -0.06 x_{14} & +  0.03 x_{12} & -0.97 x_{5} & +  0.05 x_{1} & -0.04 x_{6}\\
 x_{4}   &  3.91477272727 & -0.06 x_{14} & -0.24 x_{12} & +  1.76 x_{5} & +  0.68 x_{1} & -0.22 x_{6}\\
\hline
z    &  6.60227272727 & -0.62 x_{14} & -0.11 x_{12} & -3.11 x_{5} & -1.82 x_{1} & -0.03 x_{6}\\
\end{array}\]
Final Dictionary
Solution:  6.60227272727
Num Pivots:  4
\end{document}
